Does anyone have any decent guides on Dice Probabilities? Looking to improve my gameplay and thinking a guide on dice probabilities might help!?!?

Also in one small article ( buggered if I can find it now) it went on to say that in a "stalemate" scenario where both players are just reinforcing every turn on a particular border equally, that the more being stacked means whoever "blinks first" and attacks has more advantage. That attacking with 100 vs 100 is more beneficial to the attacker than say 10 vs 10.

My guess at your second question is that the bigger the armies the more likely probability will average out. Small armies run the risk of getting a string of losses. When you get down to 5 vs 5 or something you don't have enough of an advantage to guarantee a win in spite of one or two losses. To win in a 5 vs 5 scenario you can't afford to lose more than one roll or you lose your dice advantage. Maybe this holds true up to 10 vs 10?

http://www.plainsboro.com/~lemke/risk/ wrote: Sample interpretation of the last data above (three vs. two). If an attacker starts with 1000 armies and a defender starts with 1000 armies and a 3 vs. 2 attack is ensued, the results should be (given fair dice): after 100 rolls, each side will have lost 1 army about 34 times. The defender will have lost 2 armies about 37 times, and the attacker will have lost 2 armies 29 times. Therefore, after 100 rolls, the attacker should have 908 armies left, and the defender should have 892 armies left.

Conclusion: heads up with three dice versus 2 dice, the attacker has an advantage in the long run. Similar interpretations can be made for the remainder of the data, which can be summarized as follows:

Attacker 1 versus defender 1: defender has the advantage, winning about 4 out of 7 battles Attacker 2 versus defender 1: attacker has the advantage, winning about 4 out of 7 battles Attacker 3 versus defender 1: attacker has the advantage, winning about 2 out of 3 battles Attacker 1 versus defender 2: defender has the advantage, winning about 3 out of 4 battles Attacker 2 versus defender 2: defender has the advantage, winning about 3 out of 5 battles Attacker 3 versus defender 2: attacker has the advantage, but the advantage is much more narrow than any of the battles described above. The attacker's advantage is such that he will win about 7 out of 13 battles on average.

rockfist wrote:Dice odds are important, but many inexperienced players should take note of this:

Sometimes your best chance of winning a game is to proceed with an attack where the odds don't favor victory.

Can you give some examples where this would be applicable?

I sometimes do it myself to be honest.

If I'm in a situation where the longer the game goes the more advantage the other player has, say we're in a stale mate situation and he's reinforcing 11 per turn and me only 5 then yeah attacking sooner is beneficial because as time goes on he's going to be getting more and mroe troops compared to me so even though I might lose the battle it's still better odds than if I wait some time.

Also if stuck down in Australia and it's a fairly even gamethen and I have enough troops defending my borders I'll send out a small army every few turns just in case I get some good luck. In the grand sceme of things if they the small army dies then I've not lose too much but if I get lucky then it can quickly change the game to my favor.

Also in one small article ( buggered if I can find it now) it went on to say that in a "stalemate" scenario where both players are just reinforcing every turn on a particular border equally, that the more being stacked means whoever "blinks first" and attacks has more advantage. That attacking with 100 vs 100 is more beneficial to the attacker than say 10 vs 10.

I never understood why!

[Note]The following is based on long term law of averages and results will vary from short term attacking results.

That's because when ever 3 attacker's dice roll against 2 defender's dice, the attacker has a 54% odds of winning the roll. Now it may appear as close to 50/50 odds but what we are really looking at is close to a 10% loss for the defender over the long run (4% plus 6% equals 10%). And the more troops involved the more pronounced and definable it becomes.

The longer that the 3 attacking dice Vs. 2 defending dice is prolong the more recognizable is the advantage. 10 troops attacking 10 troops roll a maximum of 8 times with the advantage of 54% winning odds for the attacker. After that the Attacker odds are decrease by a lot. So out of 10 troops each, and after 8 rolls you can expect an average loss of (very) roughly 7.5 troops for the attacker to 8.5 troops for the defending force.

Or in other words and perhaps more correctly they both lose 8 troops each on average for every 8 rolls counted but the defender loses 1 extra troop and the attacker 1 less troop. Or The Attacker loses 7 troops to the defender's 8(+) troops lost. The end would look this. The Attacker would be left with 3 troops on the attacking region and the defender would be left with only 1+ or most likely 2 troops on the defending region (on average). At this point the odds change rather badly for the attacker.

When the 54% percent odds is then multiplied by 100 troops on the regions, then it would look something like this; 30 troops left on the Attacking region out of 100 troops for the attacker and 10+ troops left on the defending region out of 100 troops. Obviously 30 troops left over defeating 10 troops left over is better than 10 vs. 10. So the greater the troops the more pronounced and visible the 54% win ratio is. Remember that it is 54% winning odds and not 54% of the total troops. So as the number of rolls is increase so is the potential loss for the defender. (Each roll is a potential 2 troops.)

Again; 100 troops gives us a potential 99 rolls where as 10 troops gives us a potential 8 rolls at 54%. In 8 rolls the attacker loses 4 rolls or half with the rough exception of that 10% of the troops or 1 troop difference on average for every 8 rolls. With 99 rolls the potential is the same as the 8 rolls only expanded. The only thing that is carried is the analogy of 8 rolls we now use it in 80 rolls. So that both sides lose approximately 80 troops in 80 rolls but the Attacker loses 10 troops less while the defender loses 10 troops more. So that the Attacker loses only 70 troops while the Defender loses 90 troops in approximately 80 rolls (or so). So you never really get to roll 99 in 100 troops as the battle is won in some 80 rolls or so. More likely less.

I don't know how to make it any clearer as math was never really my strong suit. But perhaps there are some mathematicians out there who can either simplify this or explain it to me if I am wrong in my thinking of this exponential problem/equation.

(Revised 9/21/2012)

Last edited by Viceroy63 on Thu Sep 20, 2012 11:41 pm, edited 6 times in total.

rockfist wrote:Dice odds are important, but many inexperienced players should take note of this:

Sometimes your best chance of winning a game is to proceed with an attack where the odds don't favor victory.

Can you give some examples where this would be applicable?

I sometimes do it myself to be honest.

If I'm in a situation where the longer the game goes the more advantage the other player has, say we're in a stale mate situation and he's reinforcing 11 per turn and me only 5 then yeah attacking sooner is beneficial because as time goes on he's going to be getting more and mroe troops compared to me so even though I might lose the battle it's still better odds than if I wait some time.

Also if stuck down in Australia and it's a fairly even gamethen and I have enough troops defending my borders I'll send out a small army every few turns just in case I get some good luck. In the grand sceme of things if they the small army dies then I've not lose too much but if I get lucky then it can quickly change the game to my favor.

My record on Classic is abysmal (mainly because I play it to pick up medals) so I'm best not commenting on that specific board.

However, in general terms for instance a 6 against 7 attack is not likely to succeed, however in trench if you know your opponent will secure a bonus the next round and you need to proceed with the 6v7 attack to have any borders against his bonus area I would do it (presuming its a worthwhile bonus to oppose). Or to break someone attacking 6v7 may be worth it...or in the instance you said where they are out-deploying you by a decent clip. I'm speaking more about 1v1 games here than large games.

I agree with the above statement about Trench Warfare. Only in Trench Warfare settings is the defender given the 54% attacking odds in defense against the advancing armies. Under these circumstances it is to the advantage of the defender to assault the advancing forces regardless of if he can win or not. This makes the tactful retreat a very powerful tool. To not attack the advancing force in defense, is to give the advantage of the 54% winning odds fully to the attacking/advancing force.

And to retreat without even a partial assault is to surrender the position without a fight!